Method and apparatus for determining characteristic electrical material parameters of semi-conducting materials

ABSTRACT

The present invention relates to a method for determining characteristic electrical properties of semi-conducting materials wherein the time/frequency dependent electrical impedance or admittance of the material is measured. The invention also relates to an apparatus for carrying out the method. A number of bulb and surface parameters characterize the electrical properties of a given piece of material. These parameters include the dielectric constant ε of the material, the difference Δμ ch  in the chemical potential of the bulk of a material and the chemical potential of its surface and/or metal electrode--material surface interface, the density of the majority and minority electrical mobile charge carriers N and N min , respectively, in the bulk of the material, the electrical mobility μ of the majority electrical mobile charges in the bulk of the material and the electrical mobility μ min  of minority mobile charge carriers, the surface and bulk emission and capture rates E.R. and C.R., respectively, for mobile positive and negative charge carriers characterizing the effect of surface and bulk localized states within the band gap, when they are present, on the electrical transport.

The present invention relates to a method for determining characteristicelectrical properties of semi-conducting materials wherein the time orfrequency dependent electrical impedance or admittance is measured. Theinvention also relates to an apparatus far carrying out the method.

A number of bulk and surface parameters characterize the electricalproperties of a given piece of material. These parameters include thedielectric constant ε of the material, the difference Δμ^(ch) in thechemical potential of the bulk of a material and the chemical potentialof its surface and/or of metal electrode--material surface interface,the density of the majority and minority electrical mobile chargecarriers N and N_(min), respectively, in the bulk of the material, theelectrical mobility μ of the majority electrical mobile charges in thebulk of the material and the electrical mobility μ_(min) of minoritymobile charge carriers, the surface and bulk emission and capture ratesE.R and C.R., respectively, for mobile positive and negative chargecarriers characterizing the effect of surface and bulk localized stateswithin the band gap, when they are present, on the electrical transport.

Depending on the experimental arrangement the present invention can beused either to characterize the whole piece of material (integralcharacterization), or alternatively to map the variations of the abovementioned electrical material parameters, i.e. some of them, all of themand/or various combinations of them such as the specific resistivity ρ:1/N.e.μ as a function of the position of the measuring head andmeasuring point on the surface of the material be that a piece ofsemi-conductor, insulator or a semi-conducting wafer.

There exists a number of experimental methods today that are being usedfor the experimental determination of the above mentioned electricalmaterial parameters Δμ^(ch), ε, N and μ, in semi-conductors andinsulators. Table 1 illustrates to the best knowledge of the inventorsthe range of the experimental methods (the list not being exhaustive)that are relevant to the present invention.

The literature about each of these experimental methods is vast. Thereferences (1), (2) and (3) relate to Standard Electrical ImpedanceExperimental method and have been chosen as sufficient backgroundreference since the present invention relates directly to thisparticular experimental method.

The known experimental methods used today for the determination of theparameters suffer from a number of disadvantages the main of which canbe summarized as follows:

a) None of the existing experimental methods is capable of determiningall of the parameters simultaneously and independently in a singleexperiment.

b) A combination of two different experimental methods is often requiredin order to be able to determine some of the parameters in question, forexample a combination of the Hall effect and the d.c. four proberesistivity measurements is required if both μ and N are to be obtained.

c) Determination of some of the parameters in question is often veryindirect. The determination of Δμ^(ch) by Photoemission, for example, ismeasured in two steps. First the work function of the un-contactedsurface of the material is measured and then the measurement of the workfunction of the contacting metal is performed. The difference betweenthe two work functions is then assumed to be a measure of the Δμ^(ch)when the material in question is contacted by the given metal.

d) The experimental methods in question are often very expensive toestablish (PEE,FTIR, FTIPL), requiring either an ultra high vacuumenvironment (CP,TEE,PEE), liquid Helium temperatures (FTIR, FTIPL) ormagnetic fields (HE).

e) Because of the very nature of the experimental arrangements theexisting methods can often be considered only as qualitative in relationto the determination of the numerical values of the parameters inquestion.

f) All the electrical experimental methods named in Table 1 (StandardEIS, 4PDCR, HE and TOF) either neglect totally or treat inadequately theeffects of the electrical contact regions, the effects of the realsurfaces of the material under investigation (surfaces that are part ofthe electrical measuring circuit), the effects of the metalelectrode--material surface regions and the effects of the depletionregions within the material under investigation on the calculated valuesof the parameters in question. This neglect renders them unusable forthe characterization of higher resistivity materials (ρ≧100 Ω·cm) andintroduces relatively large errors even for characterization ofmaterials with resistivities in the region 1.0<ρ<100 Ω·cm (averaging ofthe effect of electrical contacts, depletion regions and bulk).

Among other experimental methods (not displayed in Table 1) used todayfor the characterization of semiconductors there are two that arerelevant as far as the present invention is concerned (see EP-A-0477795,DE-A-3103611, U.S. Pat. No. 5,103,183 and U.S. Pat. No. 4,028,207) andwhich should therefore be mentioned.

These are the C-V and DLTS experimental methods and variations thereofaiming at the determination of either the doping density and/or dopingdensity profiles (C-V) or at the characterization of the deep levels insemiconductors (DLTS).

The measured quantity in both methods is an appropriate capacitance andits variation with external d.c. voltage bias (C-V) or temperature(DLTS).

The major drawback of both is that the measuring frequency at which thewanted capacitance is measured is chosen arbitrarily. This fact oftenrenders them unusable.

The proper analysis of the electrical responses embodied in the presentinvention is essential in deciding on their validity. This prior art incomparison with the present invention is discussed in more detail at theend of this specification.

The above drawbacks of the prior art are eliminated by the methodaccording to the present invention which is characterized by the step ofestablishing a finite, positive (n-type) or negative (p-type) differencein the chemical potential between the inner and the surface of thematerial or between the inner of the material and the metalelectrode--material surface interface layer, thereby causing theformation of depletion regions near the electrical contacts, and byusing the effect of this established difference in the chemicalpotential on the measured electrical impedance or admittance, the stepof determining from this measured electrical impedance or admittanceone, more or all of the following electrical parameters of the material:

the difference Δμ^(ch) in chemical potential,

the dielectric constant ε,

the density N of majority mobile charge carriers,

the density N_(min) of minority mobile charge carriers,

the electrical mobility μ of majority mobile charge carriers,

the electrical mobility μ_(min) of minority mobile charge carriers,

the emission and capture rates E.R. and C.R., respectively, for mobilepositive and negative charge carriers characterizing the effect ofsurface and bulk localized states within the bind gap Eg, when they arepresent, on the electrical transport and the measured electricalimpedance, whereby the determination of the electrical parameters of thematerial is carried out by solving a system of equations for the totalcharge density ρ(x,t) consisting of the mobile negative and positivecharge densities and localized negative and positive charge densities inthe material supplemented with initial and boundary conditions, wherethe space-time development of ρ(x,t) determines the electric currentrunning in the external circuit in responce to the applied electricvoltage thereby defining the electrical complex impedance Z_(s) (ω) ofthe material.

One of the essential aspects of the present invention is the realizationthat the electrical characteristics of the surfaces of the materialand/or of metal electrode-surface interfaces, and the depletion regionswithin the material and of the bulk region of the material are allinterrelated in a calculable fashion that depends on the values of theelectrical material parameters Δμ^(ch), ε, N, μ, E.R. and C.R. thatcharacterize the material electrically. Therefore by including all theseregions in the analysis of the measurement of the electrical response ofthe material using for example Standard EIS experimental method thatleads to the determination of the complex electrical impedance Z (ω) ofthe material, the electrical material parameters of the material can bedetermined unambiguously and simultaneously in one and the samemeasurement and analysis process.

In other words the electrical characteristics of the electricalcontacts, the depletion regions and the bulk of the material are alldetermined by a few fundamental physical parameters that characterizethe electrical properties of a given piece of material--be thatsemiconductor single crystal ingot, a piece of polycrystallinesemiconductor, semiconductor wafer or a piece of insulating material.

Contrary to all the relevant existing electrical experimental methodsused today for similar type of measurements--methods that eitherneglect, average and/or treat inadequately the effects of electricalcontacts and depletion regions on the resulting measured electricalcharacteristics and which in their experimental arrangement all aim toachieve "good" ohmic contacts (the effect of the electrical contacts anddepletion regions being small and therefore often neglectable)--thepresent invention is based precisely on the opposite requirement, namelythe requirement of establishing nonohmic ("bad") electrical contacts,where the electrical characteristics of the electrical contacts anddepletion regions are apparent and are therefore easily measurable.

The experimentally observable quantity in question that contains all thenecessary information from which the calculation of the electricalmaterial parameters is possible is the above mentioned complexelectrical impedance Z (ω).

To fulfil the requirement of non-ohmic electrical contacts and therequirement of the possibility of a quantitative determination of theelectrical material parameters, a number of necessary steps has to betaken including the definition of the sample (the piece of the materialunder investigation subjected to a measurement geometry) andestablishment of the necessary conditions for the existence of thedepletion regions within the sample.

The present invention therefore relates to a method consisting of anumber of stages. These are:

1. Sample preparation

Definition of the Δμ^(ch) for the sample

2. Experimental arrangement

Definition of the sample geometry in the given electrical measurementcircuit

3. Method of measurement

Definition of the method to measure the electrical response of thesample

4. Method of analysis

Calculation of the electrical material parameters that characterize theelectrical properties of the sample.

The method according to the present invention will be somewhat differentin the two basic applications envisaged. These applications are namedhere as "research" and "on--line quality control" configurations, eachbeing characterized by somewhat different content of the steps of thepresent invention. The principle of the present invention is though thesame in both these configurations.

The "research" application of the present invention involves moredetailed studies of the electrical material parameters for example asfunctions of temperature, pressure, heat treatment, electrical d.c.voltage bias, optical excitation etc.

This type of studies require a proper treatment of the active surface ofthe material, deposition of the appropriate metal electrodes on suchprepared surfaces and the experimental environment (arrangement) thatfulfils the above mentioned requirements concerning the externalparameters, such as temperature, pressure etc. If an x,y mapping isrequired, this will be done destructively by cutting a given piece ofmaterial (semiconductor wafer) into a number of elements (x,y) andmeasuring these sequentially.

The "on-line quality control" application of the present invention onthe other hand is intended for the quality control (integral or x,ymapping) of semiconductor wafers. The electrical contacts at a given x,yposition have to be made quickly (non-destructively) and after themeasurement the measuring head (point) has to move reasonably quickly toa new x,y position, so that x,y mapping of the wafer of one or more ofthe electrical material parameters can be made reliably and fast andwith sufficient throughput (number of wafers processed in unit of time).This application of the present invention therefore involves no or verylittle sample preparation (production wafers) and fast, automated sampleexchange mechanism together with x,y scanning of the measuring headacross the surface of the wafer.

The apparatus for carrying out the method comprises an externalelectrical circuit for measuring the time or frequency dependentelectrical impedance or admittance of the material and means forelectrically contacting the surface or surfaces of the material and theapparatus is characterized by means for providing a finite, positive(n-type) or negative (p-type) difference in the chemical potentialbetween the inner and the surface of the material or between the innerof the material and the metal electrode--material surface interfacelayer, thereby causing the formation of depletion regions near theelectrical contacts, and means for determining from the measuredelectrical impedance or admittance one, more or all of the followingelectrical parameters of the material:

the difference Δμ^(ch) in chemical potential,

the dielectric constant ε,

the density N of majority mobile charge carriers,

the density of N_(min) of minority mobile charge carriers,

the electrical mobility μ of majority mobile charge carriers,

the electrical mobility μ_(min) of minority mobile charge carriers,

the emission and capture rates E.R and C.R., respectively, of mobilepositive and negative charge carriers characterizing the effect ofsurface and bulk localized states within the band gap Eg, when they arepresent, on the electrical transport and the measured electricalimpedance.

The invention will now be described in more details with reference tothe drawings in which

FIG. 1 is a schematic diagram of the one-electron energy levels in thebulk of the material, at the surface of the material and in the metalelectrode that are relevant to the present invention. (Definition of theΔμ^(ch))

FIG. 2a, 2b, 2c are schematic diagrams of possible capacitive couplingsbetween the material and the external electrical field (externalelectrical metal electrodes) that may be utilized to implement thepresent invention.

2a) "Research" configuration ("sandwich" capacitive coupling)

2b) "on-line quality control" configuration ("sandwich" capacitivecoupling)

2c) "Research" and/or "on-line quality control" configurations ("planar"capacitive coupling) (Definition of the sample geometry)

FIG. 2d,2e are schematic diagrams showing the geometrical dimensions ofthe material under investigation

2d) "Research" configuration

2e) "on-line quality control" configuration (Definition of the samplegeometry)

FIG. 3 is a schematic diagram of the experimental measuring arrangementwhich may be utilized to measure the electrical complex impedance Z (ω)of the series combination--sample bulk, sample depletion regions andsample electrical constant regions (sample surface-metal electrodeinterface). (Definition of the measurement method)

FIG. 4a, 4b,4c, 4d,4e is a schematic diagram of the passive R,Celectrical network model that approximates the electrical response ofthe material (bulk, depletion and electrical contact regions) to a smallexternal electrical field input signal. (Definition of the approximativeR,C model for the electrical response function Z_(s) (ω) of the sample)

FIG. 5 is a schematic graphical representation of the electricalresponse function Z (ω), (Y(ω)) of the material sample modelled by FIG.4d, comprising the bulk, the depletion and electrical contact regions ofthe sample.

FIG. 6a, 6b show Example 1; the experimentally observed electricalresponse of the sample 1 (ultra pure silicon monocrystal as a functionof temperature T. (ω=2π·f)

6a) The logarithm of the real part of the electrical impedance Z (ω)versus logarithm of the measuring frequency ω,

6b) The logarithm of C (imaginary part of the electrical admittance Y(ω)divided by (ω) versus logarithm of measuring frequency ω.(Results--example 1)

FIG. 7a, 7b show Example 2; the experimentally observed electricalresponse of the sample 2 (pure, polycrystalline silicon, unknown type,unknown N) (ω=2π·f)

7a) The logarithm of the real part of the electrical impedance Z (ω)versus logarithm of the measuring frequency ω.

7b) The logarithm of C(ω) (imaginary part of the electrical admittanceY(ω) divided by ω) versus logarithm of the measuring frequency ω.(Results--example 2)

FIG. 8a, 8b, 8c The theoretical analysis of the electrical response ofthe pure, polycrystalline silicon sample (Example 2)

8a) A topological model of polycrystalline silicon sample consisting ofN₁ ×N₂ grains, each grain being characterized by its electricalimpedance Z_(j) ·(ω). The shown passive, static R,C electrical networkrepresenting the total electrical response includes alsometal-semiconductor interface region represented by R_(M+S), C_(M+S)elements.

8b) A simplified physical model of depleted grain "j".

8c) Static R,C electrical network representing the electrical responseof a single grain.

The present invention is not limited to any particular apparatus, samplepreparation and measurement method but may be carried out by a varietyof different steps (method stages).

In which follows the individual stages

1) Sample preparation (definition of the Δμ^(ch) for agiven material(sample))

2) Experimental arrangement (definition of the sample geometry withinthe given electrical measurement circuit)

3) Method of measurement (measurement of the electrical response)

4) Method of analysis (calculation of the electrical material parametersΔμ^(ch), ε, N and μ (here also all possible combinations of these) byusing various analytic expressions for parts or the whole of themeasured Z (ω) function or by using the numerical solutions of thegoverning equations with the electrical material parameters as freefitting parameters

will be described in detail using FIGS. 1 to 5. The new method forelectrical characterization of semiconducting/insulating materials willbe demonstrated on a couple of examples shown in FIGS. 6 to 8.

Finally Table 2 illustrates the quantitative results for the values ofΔμ^(ch), ε, N and μ on a number of monocrystalline silicon samples ofboth n and p-type with various degrees of doping, obtained by the use ofthe present invention and compared with the values for these parametersobtained by other methods on the same samples and from Physics tables(4) and the literature.

STAGE 1 Sample Preparation

(definition of Δμ^(ch) =μ^(ch) _(B) -μ^(ch) _(M+S) of the material)

In order to achieve the first requirement (establishment of thecondition Δμ^(ch) >0 for n-type and Δμ^(ch) <0 for p-type material,respectively) reference is made to FIG. 1 for a sample sandwichedbetween two metal electrodes in a preferred geometric configurationdepicted in FIG. 2a (a simple capacitor configuration). The problem inthis case is quasi one-dimensional (area A_(r) of the sample is constantand its linear dimension √A is large compared with the length of thesample L).

Referring to FIG. 1 the sample of the finite length L extends from x=Oto x=L, x=O and x=L planes being the true surfaces of the sample. The.metal electrodes are deposited Δx→0 (or applied Δx>0) on both thesesurfaces (capacitive coupling of the sample to the external field) andwhen an external electrical voltage is applied between these twoelectrodes an external electrical field is created within the volume ofthe sample.

With no external electrical voltage applied the average energy of themobile electrical charges well within the bulk of the sample ischaracterized by the thermodynamical quantity μ_(B) ^(ch) --chemicalpotential of the bulk. The average energy of the electrical charges atthe surface of the sample (and/or of sample surface-metal electrodeinterface) is characterized by the sample surface-metal electrodeinterface chemical potential μ^(ch) _(M+) _(S).

Considering the n-type material first there will be an outflow (inflow)of the mobile electrical charges from (into) the sample to (from) samplesurface-metal electrode interface region depending on whether thechemical potential μ_(B) ^(ch) of the bulk of the sample lies above(below) the value of the chemical potential of the sample surface-metalelectrode interface region μ^(ch) _(M+) _(S).

This flow of the mobile electrical charges from the sample interior(sample exterior) creates an internal electrical field in the regionsnear the surfaces of the sample that eventually stops any further flow.In this way a dynamical thermodynamical equilibrium is established thatis characterized by the existence of the depletion regions (outflow ofthe mobile electrical charges from the interior of the sample into thesample surface-metal electrode interface region or accumulation reqions(inflow of the mobile electrical charges from the sample surface-metalelectrode interface region into the interior of the sample).

The requirement of the Stage 1 of the present invention (Δμ^(ch) >0(n-type material), Δμ^(ch) <0 (p-type material)) is simply therequirement for establishing (securing the existence of) the depletionregions in the material by securing that for the n-type material μ^(ch)_(B) >μ^(ch) _(M+S) (the same arguments apply for the p-type materialbut here the condition μ^(ch) _(M+S) >μ^(ch) _(B)).

If the condition μ^(ch) _(B) >μ^(ch) _(M+S) (n-type material) or μ^(ch)_(M+S) >μ^(ch) _(B) (p-type material) are not satisfied then anappropriate treatment of the sample surface might be necessary. Thisapplies in cases where μ^(ch) _(M+S) is determined mainly by the surfacestates of the material. Alternatively an adequate metal electrode--withthe right value of the work function--has to be chosen. This applies incases where μ^(ch) _(M+S) is determined mainly by the work function ofthe metal electrode. In general a combination of both processes might berequired.

Both of these preparation processes or steps move the value of μ^(ch)_(M+S) up or down on the energy scale (see FIG. 1) so that the conditionΔμ^(ch) >0 can be eventually obtained for n-type and p-type materials(Δμ^(ch) <0).

Although a specific description of the Stage 1 has just been given, itis clear to anyone experienced in the field that other ways of securingthe condition Δμ^(ch) >0 might be possible.

The Stage 1 of the present invention therefore does not relate to anyparticular process or sequence of processes how to achieve the conditionΔμ^(ch) >0 but it is simply embodied in the conditionitself--establishment of the Δμ^(ch) >0 for n-type material and Δμ^(ch)<0 for p-type material.

STAGE 2 Experimental Arrangement

(definition of the sample geometry, of the sample environment and of theelectrical measurement circuit) Experimental arrangement thatconstitutes the Stage 2 of the present invention is defined in threesteps. The first is sample--metal electrodes configuration, the secondis definition of the sample geometry in a given sample - metal electrodeconfiguration and the third step defines the sample environment togetherwith electrical connections to the external electrical measurementcircuit. These steps will now be described in more detail.

Sample--metal electrodes configuration defines the way the externalelectrical field is coupled to the piece of the material underinvestigation. This can be done in a variety of ways in principle but inthe preferred embodiment (preferred experimental arrangement) a mostsimple capacitive coupling is preferred where the material and theelectrodes form a simple capacitor.

In this configuration, known also as "sandwich configuration", theelectrical field profile within the sample is simple, making thedefinition of the sample geometry in the electrical measurement circuiteasy and precise.

Two basic preferred configurations are considered. They are both shownin FIGS. 2a and 2b.

In the "research" configuration (FIG. 2a) the appropriate metalelectrodes are deposited directly onto both surfaces of the material(Δx→0 in FIG. 1) minimizing the "edge" effects of this simple capacitor(no corrections for the "edge" effects in the calculation of the samplegeometry is needed).

In the "on-line quality control" configuration (FIG. 2b) the majorrequirement is that the electrodes can be applied at a particularmeasurement point P₁ (x₁,Y₁) of the semiconductor wafer surface andafter the measurement of the electrical response is completed at thispoint the electrodes can be re-applied at a new point P₂ (x₂,y₂) with anew measurement of the electrical response to follow. The process isrepeated for all the points P_(i) (x_(i),y_(i)) of the wafer surface.The effective electrode area (including the "edge" effects) is requiredto be the same through-out the whole of the mapping measurement process(measurement of the electrical response across the whole area of thewafer).

Another possible sample--electrode configuration is shown in FIG. 2c.This type of configuration--an "open capacitor" configuration (alsoknown as "planar" configuration) might be relevant for the measurementson thin films and/or semiconducting/insulating bulk materials (here alsowafers) where a simple capacitor configuration is not possible. Thepreferred shape of the electrodes in this case will be rectangular withwell defined gap between them.

In all the configurations described above the geometrical shape of theelectrodes is not important but must be known together with thecorresponding external electrical field profile.

Sample Geometry

In a given sample--electrode configuration, either a preferredembodiment--FIG. 2a and 2b or any modification thereof such as shown forexample in FIG. 2c, the next step in Stage 2 of the present inventionconsists of the determination of the geometrical dimensions of thesystem sample+metal electrodes that are relevant for the definition ofthe given configuration as a electrical element within the electricalmeasurement circuit. The quantities in question are the areas and thelengths (thicknesses) of the individual elements comprising thesample--electrode configuration.

Referring to FIG. 2d (the "research" configuration), the elements of theconfiguration are the top metal electrode (of area A_(r) and length t₁),the piece of the material under investigation (of area A_(r) and lengthL) and the bottom metal electrode (of area A_(r) and length t2). If theelectrical resistance of the metal electrodes is low (low compared tothe resistance of the piece of the material under investigation), theeffect of the metal electrode elements on the electrical response can be(safely) neglected and the geometrical dimensions of thesample--electrode configuration is that of the sample itself (area A_(r)and the length L).

In the "on-line quality control" configuration shown in FIG. 2e the samearguments apply and the relevant elements (with their geometricaldimensions) are the contacting conductive medium (of area A₁ and thelength Δ x; in the preferred embodiment of the present invention thesequantities are the same for both electrical contacts), the piece of thematerial under investigation (of the effective area A₂ (inclusive the"edge" effects) and the length L) and the top and bottom metalelectrodes (both of the area A₃ and the length t).

All these quantities have to be measured with a sufficient precisionprior to calculation of the four electrical material parameters Δμ^(ch),ε, N and μ since the relative errors in their measurement determines theoverall accuracy with which the numerical values of the above mentionedfour electrical material parameters can be obtained using the presentinvention.

Sample environment and electrical contacts

Two basic sample environments are envisaged in the preferred embodimentof the present invention.

In the "research" configuration the sample (FIG. 2d) is placed in thecryostat, so that the dependence of the four electrical materialparameters Δμ^(ch), ε, N and μ can be studied as function oftemperature, pressure, d.c. electrical bias, optical (particle)excitation of the electrical charges within the material underinvestigation etc.

In the "on-line quality control" configuration, the sample (asemiconductor wafer--see FIG. 2e) is placed preferably in a automaticwafer exchange mechanism that introduces one wafer at a time into a dark(optically) measurement compartment, where x,y mapping of the fourelectrical material parameters takes place at room temperature, thewhole assembly making the large throughput (number of wafers processedin a unit of time) possible. Other sample environments are evidentlyalso possible and this step of the Stage 2 of the present invention doesnot therefore relate to any particular sample environment but onlydemands that the sample environment be well known (temperature of thesample, definition of the "optically" dark measurement compartmentetc.).

Once placed in its appropriate environment the sample has to beconnected via electrical leads to the external measurement circuit. Inthe preferred embodiment of the present invention this is achieved bestby connecting four screened coaxial lines (two and two connectedtogether at the sample--see FIGS. 2a-2c) to each of the sample surfaces(each of the sample electrodes), minimizing in this way the unwantedeffects of the external leads (the self-induction and the straycapacitances of the electrical leads) on the measured electricalresponse of the sample. Two of the leads (one from each side of thesample) are used to measure the electrical voltage across the samplewhile the other two leads are used to measure the electrical currentthrough the sample. The other ends of the four coaxial lines are thenconnected to the external electrical measurement circuit.

Other possible ways of connecting the sample to the external electricalmeasurement circuit exist evidently also.

This step of the Stage 2 of the present invention (electrical contactsto the sample) does not relate to any particular way of connecting thesample to the external electrical measurement circuit but resides (isembodied) in the requirement that the effect of the external electricalleads to the sample in the alternatives (modifications) to the preferredembodiment (FIG. 2a and FIG. 2b) must be known and properly accountedfor in the final analysis of the electrical response of the wholesystem, the system being defined as consisting of the sample, theelectrodes and the electrical leads.

Stage 3 Method of Measurement

(definition of the measurement method)

There exists a number of possible ways of measuring the electricalresponse of a system (sample, electrical metal contacts and externalelectrical leads) involving either real time and/or frequency domain.All these experimental methods involve a measurement of the two relevantquantities--input signal (be that electrical charge, electrical voltageor electrical current) and the--output signal (complementary quantity tothe input), the ratio between the output and the input defining theappropriate electrical response function. Two forms of the input signalare used most commonly. These are the "step" or "sinusoidal" input. Therespective electrical response functions of these two input forms arerelated to each other through Laplace transform (linear responseregime).

A typical electrical measurement arrangement is shown schematically inFIG. 3. In the preferred embodiment of the present invention asinusoidal voltage from the voltage source is applied across the sample(voltage input) and its amplitude is measured by a "vector" voltmeter.The amplitude and the phase shift (relative to the voltage input) of theoutput current (electrical current measured in the external electricalcircuit) is measured by a "vector" ammeter and the ratio of the in-phaseand out-of-phase component of the current output to the voltage inputthen defines the complex electrical admittance Y (ω) of the system underinvestigation (system being defined as sample+electrodes+externalelectrical leads).

Another quantity that characterizes the electrical response of thesystem under investigation is the complex electrical impedance of thesystem defined as Z (ω)=1/Y (ω).

The direct experimental determination of the Z (ω) is accomplished inpractice most conveniently by various types of low frequency (ω≦1.10¹⁰Hz) impedance a.c. bridges consisting of a voltage source with avariable frequency f (ω=2IIf), preferably with a range of 1 mHz through10 GHz, "vector" voltmeter, "vector" ammeter and some additionalcircuitry to yield the function Z (ω) at various frequencies of theinput sinusoidal voltage. All these components form elements of a singlemeasurement unit shown schematically in FIG. 2a and FIG. 2b.

The Stage 3 of the present invention does not depend on any particularway of the measurement of the electrical response of the system underinvestigation but relates to a process consisting of the measurement ofthe electrical response of the system and its subsequent representationin the frequency domain by the complex electrical impedance (admittance)Z (ω) (Y (ω)). This representation is not (strictly) necessary and otherrepresentations of the electrical response of the system underinvestigation, such as for example the complex dielectric constant ε(ω), the complex electrical conductivity σ (ω), the complex electricalmodulus M (ω) and/or various variations of the electrical responsefunctions in the real time domain, are all of course possible. The Z (ω)(Y (ω)) representation of the electrical response of the system ispreferred for the reasons of clarity of the physical arguments behindthe analysis of the electrical response data and because the numericalvalues of these functions at various frequencies have a very definitephysical meaning in relation to the four electrical material parametersΔμ^(ch) ε, N and μ that are to be extracted from such an analysis.

Within the necessary experimental range of frequencies f (1 mHz-10 GHz)will these numeric values fall within the following typical ranges formost of studied semiconductors/insulators:

the real part of impedance in the range from 0.00t ohm to 10¹⁴ ohm. theimaginary part of admittance divided by ω from 10⁻¹⁴ Farad to 10⁻²Farad.

Stage 4 Method of Analysis

(calculation of the electrical material parameters Δμ^(ch), ε, N an μ ofa material under investigation).

Introduction

It is at stage 4 that the known physical theories and their mathematicalformulation are applied to a specific technical problem, namely theproblem of the electrical characterization of semiconducting and/orinsulating materials in general ("research" configuration of the presentinvention) and in a more specific application, the problem or theelectrical characterization of semiconductor wafers ("on-line qualitycontrol" configuration of the present invention).

The discovery aspect of the present invention can be summarized in twomajor points:

Firstly the discovery of a direct link (that can be formulated simplyand clearly through mathematical equations and/or expressions) betweenthe full theoretical analysis of the electrical response of a material,including the depletion regions and material surface-metal electrodeinterface regions and the experimentally observable electrical responsefunction Z (ω) of the material (where the function Z (ω) describes thedependence of the complex electrical impedance of the material on thefrequency of the applied external voltage input).

Secondly the discovery of a possibility of extracting the fourelectrical material parameters Δμ^(hu) ch, ε, N and μ of the material(and/or their combinations) when the depletion and materialsurface-metal electrode interface regions are included in such atheoretical analysis.

The invention itself is then embodied in the application of thisdiscovery aspect to the technical problem of characterizing theelectrical properties of semiconducting/insulating materials andsemiconductor wafers through the determination of the four electricalmaterial parameters Δμ^(ch), ε, N and μ using the above mentioned methodof analysis of the experimentally observed function Z (ω). One of theclaims of the present invention will be therefore the use of the presentinvention to determine one, two, three, four and/or any combination ofthe four electrical material parameters to characterize electrically apiece of a material.

Because of the importance of the Stage 4 of the present invention a moredetailed description of this stage will now be given.

Firstly, the definition of the problem will be presented.

Secondly, the description of the essential parts of the theory behindthe full theoretical analysis of the electrical response of a realmaterial (material with depletion regions, surface and materialsurface-metal electrode interface regions) will follow.

Thirdly, the various possible approximations and the correspondingsolutions will be described and the link between the governing equations(and the corresponding solutions thereof) and the experimentallyobservable response function Z (ω) will be pointed out.

Fourthly the various methods of calculating the numerical values of thefour electrical material parameters Δμ^(ch), ε, N and μ will bedescribed and the validity of the method of analysis of the presentinvention will be demonstrated on a set of experimental results obtainedby the use of the present invention on a number of monocrystalline andpolycrystalline silicon samples of both types (n-type and p-type) andwith varying degrees of doping.

These results (the obtained numerical values of the four electricalmaterial parameters Δμ^(ch), N and μ) will be then contrasted in thefinal Table 2 with the results for the same four electrical materialparameters obtained either on the same samples by the other experimentalmethods or by comparing the obtained values with those in the literatureand/or in Physical Tables of material constants (4).

Definition of the problem.

Referring to FIGS. 2a, 2b and 3 the measured electrical response Z (ω)is due to the electrical response of the sample (bulk region, depletionregions and the surface-metal electrode interface regions), theelectrical response of the contacting, conductive medium (FIG. 2b), theelectrical response of the metal electrodes (FIGS. 2a and 2b), theelectrical response of the electrical contacts (FIGS. 2a and 2b) and theelectrical response of the external electrical leads (FIGS. 2a and 2b)connecting the sample configuration within a given sample environment tothe external measurement circuit.

In order to obtain the electrical response function Z_(s) (ω) of thesample alone, it is therefore necessary to subtract from the totalelectrical response function Z_(ToT) (ω) of the total system theelectrical response from all the unwanted elements of the system(contacting, conductive medium (if present), metal electrodes,electrical contacts and the external electrical leads.

This subtraction is in principle easy since the elements of the systemare connected in series so that the total electrical response functionZ_(ToT) (ω) of the system is the sum of the complex electricalimpedances of the individual elements of the system. Provided that theelectrical impedances Z_(i) (ω) of the individual elements of the systemare known (this can be rather difficult to achieve in practice), thissubtraction can be performed.

In the preferred embodiment of the present invention, the effect ofZ_(i) (ω) of the elements of the system on the resulting totalelectrical response function Z_(ToT) (ω) is minimized, so that to a verygood degree of precision (typically better than 1% accuracy) theseeffects can be neglected and the measured total electrical responsefunction Z_(ToT) (ω) of the system represents the true electricalresponse of the sample Z_(s) (ω).

Having extracted the electrical response function Z_(s) (ω) of thesample from the experimentally observed Z_(ToT) (ω) (if modifications tothe preferred embodiment of the present invention are used), the nextstep is to subject this function to the theoretical analysis embodied inthe present invention.

The application of this analysis then results in the possibility of thedetermination of the numerical values for the four electrical materialparameters Δμ^(ch), ε, N and μ (and/or their combinations such as forexample the specific resistivity of the bulk region of the material)that characterize the electrical properties of the material(semiconductor, insulator, semoconductor wafer etc.)

The results of the present analysis can be summarized very schematicallyas follows:

By including depletion regions and the surface (and/or materialsurface-metal electrode interface) regions of the material underinvestigation in the full analysis of the electrical response of thesample (rather than to try to avoid it at all costs, as has been thecase up till now), the analysis method of the present invention showsthat the form or the values of Z_(s) (ω) in various frequency regionsand/or at various frequencies are related directly to the fourelectrical material parameters Δμ^(ch), ε, N and μ of the material.

This means for example that under appropriate experimental conditionsthe value of the real part of the Z_(s) (ω) at high frequencies isrelated directly to the bulk resistivity ρ of the material (ρ: 1/N·e·μ),while the imaginary part of the Y_(s) (ω) divided by ω is relateddirectly to the bulk dielectric constant ε.

Similarly, at low frequencies both the real part of Z_(s) (ω) and theimaginary part of Y_(s) (ω) divided by ω are directly related to theparameters Δμ^(ch) and N.

Furthermore, by looking at the form of Z_(s) (ω) (and/or Y_(s) (ω)) inthe appropriate frequency regions it is possible to establish the typeof the majority charge carriers and whether the electrical transportwithin the material .is due to only one kind of mobile electricalcharges (electrons or holes) or whether both mobile electrical charges(electrons and holes) contribute significantly.

Finally, it is possible for the first time to study quantitatively thebulk region, the depletion regions and the surfaces (and/or materialsurface-metal electrode interface regions) of the material underinvestigation independently and separately by measuring (studying) theelectrical response function Z_(s) (ω) in various frequency regions.

Description of the theory behind the analysis method (the elements off)

a) Electrical response in solids at classical frequencies

A new analysis of the electrical response of a macroscopic solid statesystem to an arbitrary electrical voltage input indicates that with asmall sinusoidal voltage input within the classical frequency range(ω<10¹⁰ Hz), the electrical response in a solid state system mirrorsfirst of all the static, spatial distribution of the mobile chargecarrier density through the local electrical conductivity relaxationtime τ.sub.σ (x)=ε(x)/σ_(D).C. (x) (ε(x) is the local dielectricconstant, σ_(D).C. (x) is the local d.c. electrical conductivity) (seeFIGS. 4b and 4c).

This enables any system, inclusive the depletion regions and theelectrical contact regions of a material to be modelled by a simple,passive R,C electrical network, where each of the network elements has avery direct physical meaning.

Furthermore, since the electrical conductivity relaxation time is moreor less the only relevant characteristic time of the problem, theregions of the sample with different τ.sub.σ will respond at differentcharacteristic frequencies that are directly related to τ.sub.σ. In thisway it is therefore possible to study for example the bulk, thedepletion and the electrical contact regions separately by studying theelectrical response of the sample (system) in different frequencyregions.

b) Mathematical formulation

The macroscopic electrical response of a medium (characterized by adielectric constant ε and the electrical mobility μ) at classicalfrequencies is described fully by classical electrodynamics (two ofMaxwell equations), the constitutive equation defining the total localelectrical current, the initial condition for the total, electricalcharge density ρ(x,t) spatial distribution and by the boundary conditiondefining the charge transport across the boundaries.

Within the classical range of frequencies (ω<10¹⁰ Hz) the dielectricconstant ε of the material and the electrical mobility μ of the mobileelectrical charges within the (bulk of) the material can be bothconsidered as time independent (possible slow polarisations neglected inthe first approximation).

In one dimension (the appropriate experimental arrangement) and whenboth types of the mobile charge carriers (electrons at E_(c) and holesat E_(v) --see FIG. 1) contribute to the transport, the definingequations can be re-cast into a set of coupled, non-linear parabolicequations for each type of the mobile charge carrier particle density:##EQU1##

The constitutive equation defining the total local electrical currenthas the form ##EQU2## and the boundary condition for ρ(x,t) is definedthrough the relation ##EQU3##

Here μ_(e) and μ_(h) are the electrical mobilities of the electrons atE_(c) and of the holes. at E_(v) respectively, ε is the dielectricconstant of the material, k is Boltzmann constant, T is the absolutetemperature and |e| is the elemental electrical charge.

The quantities n_(c) (E_(c),x,t), n_(h) (E_(v), x,t), E (x,t) and ρ(x,t)are the particle number densities of the respective mobile chargecarriers, the local electrical field and the local, total charge densityrespectively. N_(i).LOC⁺ (E_(i),x,t) and N_(j).LOC⁻ (E_(j),x,t) are thelocal particle number densities of charged, localised energy levelsE_(i) and E_(j) within the energy band gap E_(g).

(E.R.)_(i), (C.R.)_(i), (E.R.)_(j) and (C.R.)_(j) are their respectiveemission and capture rates and finally G_(e), R_(e) and G_(h), R_(h) arethe generation and recombination rates for mobile electrons at E_(c) andholes at E_(v) respectively.

c) Results--Dynamical solution

A set of the governing equations (1) and (2) together with theappropriate initial and boundary conditions for n_(e) (E_(c),x,t), n_(h)(Ev,x,t) (mobile electron and hole particle densities), N³⁰ _(i),LOC(E_(i),x,t) and N⁻ _(j), LOC (E_(j),x,t) (localized positive andnegative charge particle number densities) then determine the space-timeevolution of the total charge density ρ(x,t) and therefore also theelectrical current flowing in the external circuit in a response to agiven applied voltage input, thereby defining the electrical compleximpedance of the sample Z_(s) (ω).

d) Results--Long time, static limit

For times t>>0 (t→∞) and with no external applied field, the set of thegoverning equations leads to the formation of the (majority) mobilecharge carrier depleted regions near the surfaces if the boundarycondition reflects the finite difference in the chemical potentialacross the boundary at time t=0. In this case the space-time evolutionof the total charge density ρ(x,t) will approach the equilibrium, timeindependent distribution as the time goes to "infinity" t_(o) →∞. Thisdistribution is also a solution of the static Poisson equation to whichthe problem reduces in this long time, static limit.

e) Results--Small signal approximation and the static R,C networksolution to the problem of the electrical response

One of the major results of the present dynamical analysis of theelectrical response in solids at classical frequencies is concerned witha small input signal approximation.

When the external applied voltage is sufficiently small, theequilibrium, spatially non-homogeneous distribution of the mobile chargecarrier particle densities will not be disturbed by the applied externalelectrical field and it can be shown that the electrical response of theentire system (sample) under these conditions is identical to a responsefrom simple, parallel R,C electrical elements, connected in series.Contrary to the usual passive R,C network models of various junctions,interfaces and semiconductor--insulator--metal structures, theelectrical elements in the present static R,C network have a very directphysical meaning and are all interrelated.

According to the result of the present analysis, the sample is simplydivided into a number of volume elements (see FIGS. 4a, 4b and 4c), theactual number depending on the required precision with which theelectrical response is required. Each volume element ΔV_(j) ischaracterized by its electrical resistance R_(j) (in-phase component ofthe response; usually dissipation of the external field energy throughfinite electrical mobility μ of the material) and by its geometricalcapacitance Cj (out-of phase component of the response; fastpolarisation through finite dielectric constant ε of the material).

The solution of the Poisson equation in the long time, static limitdetermines the spatial distribution of the electrical mobile chargedensity within the sample and therefore also the electrical resistanceof each of the volume elements ΔV (x_(j)). The geometrical dimensions ofeach volume element then determine its characteristic geometricalcapacitance.

By defining R's and C's for all the volume elements comprising thesample (bulk region), the depletion regions and surface-metal electrodeinterface regions) one has determined also the total electrical compleximpedance of the sample Z_(s) (ω) which is simply the sum of theimpedance of the individual volume elements.

As can be seen from FIG. 4c each of the volume elements has got its owncharacteristic electrical conductivity relaxation time τ_(j) and the"visibility" of the depletion regions, bulk region and the contactregions of the sample in the xperimentally observed Z_(s) (ω) becomesnow obvious. Depending on the frequency of the applied externalelectrical field (applied external voltage input), the different partsof the sample will be (or will not be) able to respond.

In the long time, static approximation therefore, the measurement of theelectrical response (Z_(s) (ω) at different frequencies is directlyrelated to the equilibrium spatial distribution of the mobile electricalcharge particle density in the different parts of the sample.

Identification of the link (correlation) between the theoreticalanalysis of the electrical response and the experientally observedelectrical response function Z_(s) (ω).

It is the discovery of a direct correlation between the space-timeevolution of the total charge density ρ(x,t) within the sample and theexperimentally observable electrical impedance of the sample Z_(s) (ω)together with the solutions for ρ(x,t) (either a dynamical solutionobtained by solving the whole set of governing equations numerically ora static solution obtained by solving the static, time independentPoisson equation with corresponding reticulation of the sample into thesum of the volume elements, each characterized by its own impedance),that constitutes the discovery aspect in the theoretical analysisembodied in the present invention.

The use of this discovery aspect to characterize the electricalproperties of semiconducting/insulating materials and semiconductingwafers through the extraction of the four electrical material parametersΔμ^(ch), ε, N and μ (either some of them, all of them or somecombination of them) using this method of analysis (together with theother stages described) then constitutes the present invention.

Calculation of the four electrical material parameters Δμ^(ch), ε, N andμ of the material under investigation.

Depending on the particular material under investigation, thecalculation of the electrical material parameters from theexperimentally observed electrical response function (complex electricalimpedance) of the sample Z_(s) (ω) can proceed (can be performed) atthree possible levels described in the foregoing paragraphs. Theselevels will now be described in more detail in order of increasingcomplexity.

Level 1--"Hat" model approximation.

It turns out that within the long time limit static approximation afurther simplification can be employed in the static R,C networkanalysis of the electrical response of the sample that makes thedetermination of the four electrical material parameters Δμ^(ch), ε, Nand μ even more direct. If the (characteristic) frequency regionscorresponding to the response from the material surface-metal electrodeinterface regions, from the depletion regions and from the bulk regionof the sample are all well separated, it is possible to lump all thevolume elements of the bulk region together and do the same for thedepletion regions and the sample surface-metal electrode interfaces. Theelectrical response of the sample is then represented by a very simplestatic R,C network consisting now of only three R,C elementsrepresenting in turn the electrical response from the samplesurface-metal electrode interfaces (time constant τ_(M+S)), theelectrical response from the depletion regions (time constant τ_(D)) andthe electrical response from the bulk of the sample (time constantτ_(B)). This representation (named here as "Hat" model approximation) ofthe electrical response of the sample shown in FIG. 4d is sufficientlyprecise only in case when the characteristic time constants of therespective regions of the sample are well separated (τ_(M+S) >τ_(D)>τ_(B)).

The "Hat" model reticulation of the sample shown in FIG. 4d enables oneto write down analytic expressions for the four quantities R_(D), C_(D),R_(B) and C_(B) that can be identified in the "raw" experimental data ofthe measured Z_(s) (ω) of the sample.

The expected frequency dependence of Z_(s) (ω) for a sample representedby the equivalent electrical circuit of FIG. 4d is shown in FIG. 5. Forclarity purpose the real part of Z_(s) (ω) (dimension Ohm) and imaginarypart of Y_(s) (ω)/ω (dimension Farad) are displayed. The flat parts ofthe curves denoted by symbols R₂, R₃, C₂ and C₃ are directly related toR_(D), R_(B), C_(D) and C_(B) through algebraic relations ##EQU4## whichin turn are defined in terms of the four electrical material parametersΔμ^(ch), ε, N and μ of the material under investigation.

For the case of a typical semiconductor/insulator with a band gap E_(g)and with one-particle effective density of states N_(c) (conductionband) and N_(v) (valence band) the defining expressions are ##EQU5##

The quantities appearing in the analytical expressicns have thefollowing meaning:

μ(T)--electrical mobility of the mobile, majority el. charge carriers

μ_(e) (T)--electrical mobility of the electrons at E_(c)

μ_(n) (T)--electrical mobility of the holes at E_(v)

N--mobile, majority el. charge carriers particle number

density

L--total length of the sample

W_(D) (T)--depletion width of the sample

A_(r) --total, active area of the sample

ε_(o) ·ε_(r) =ε--dielectric constant of the sample (silicon)

E_(g) --energy gap of the material

Δμ^(ch) (T)--difference in the chemical potential

N_(c), N_(v) --one-particle effective densities of states in theconduction (E_(c)) and valence (E_(v)) bands respectively

|e|--elemental electrical charge

k--Boltzmann constant

T--absolute temperature

E_(O) ^(P) --Energy difference between the top of the valence band andthe electrochemical potential at the surface of the sample

E_(o) ^(m) --Energy difference between the bottom of the conduction bandand the electrochemical potential at the surface of the sample

Z--variable in the function arctg having integration limits u and 1.0

Considering now the four electrical material parameters Δμ^(ch), ε, Nand μ as four unknown constants difined by four independent equations,it ts clear that a unique solution exists. It can found iteratively byfirst assuming some definite values (starting values) for Δμ^(ch), ε, Nand μ, then calculating the quanties R₂, R₃, C₂, C₃ and comparing theresult with the observed values of R₂, R₃, C₂ and C₃. The next step inthe iteration procedure coonsists in changing the starting values forΔμ^(ch), ε, n, μ and repeating the whole process until the requiredprecision (agreement between the calculated and measured values of R₂,R₃, C₂ and C₃ is obtained. The fitting procedure is so simple that itcan be done by hand and does not required computer. In these cases wherethe ratio of the electrical mobilities P₂ =μ_(n) /μ_(e) is alsoconsidered as an unknown parameter, a combination of the measurements onboth n-type and p-type samples is needed in order to object a uniquesolution.

Level 2 Static Approximation Using the Solution of the Poisson Equation

When a fit to the experimentally observed electrical reponse functionZ_(s) (ω) of the sample over the entire frequency domain measured in theexperiment is required for higher precision the individual levels R₂,R₃, C₂ and C₃ can not be easily identified in the raw experimental dataZ_(s) (ω), the static R,C network analysis of the electrical response ofthe sample involves the reticulation of the sample as shown in FIG. 4e.Here the depletion region volume elements have not been lumped togetherbut rather the resistances of the individual volume elements within thedepletion region have been determined through the solution of thePoisson equation for the total local charge density ρ(x,t→∞) assumingsimple parabolic bands (parabolic band approximation--see FIG. 1). Theelectrical impedance of the depletion region can then be calculatedanalytically leading to the expression (both electrons and holescontributing to the electrical transport) for Z_(D) (ω) of the form##EQU6## and where μ_(e) (T)--electrical mobility of the electrons atE_(c)

μ_(h) (T)--electrical mobility of the holes at E_(v)

N_(c), N_(v) --one-particle effective densities of states in theconduction (E_(c)) and valence (E_(v) ) bands

ε₀ ·ε_(r) =ε--dielectric constant of the sample

|e|--elemental electrical charge

E_(g) --energy gap of the material

k--Boltzmann constant

T--absolute temperature

W_(D) (T)--depletion width of the sample

Δμ^(ch) (T)--difference in the chemical patential

N--mobile, majority electrical charges particle number density

A_(r) --total, active area of the sample.

In order to extract the four electrical material parameters Δμ^(ch), ε,N and μ of the material under investigation, the same iterativeprocedure as before is used only now the fit can be best performed by acomputer using the four electrical material parameters as free fittingparameters. The computer is used mainly for the purpose of time economy,since in principle the calculation can be done by hand due to the factthat the solution for the Z_(S) (ω) is analytical also in this case. Ifthe analytical expressions for the interface element (materialsurface-metal electrode interface region) are not known, the fit to theexperimentally observed Z_(D) (ω) is performed using R_(M+S) and C_(M+S)as two extra phenomenological free fitting parameters. In case ofsilicon though the analytical expressions for the interface elementsR_(M+S), C_(M+S) do exist as will be demonstrated in the experimentalverification of the present invention. The effect of R_(M+S) and C_(M+S)beyond the frequency 1/τ_(M+S) (FIG. 5) is small, the form of theresponse function Z_(s) (ω) in this frequency region being determinedmainly by Z_(D) (ω) and the known bulk element Z_(B) (ω) (elements R_(B)and C_(B)).

Level 3 Dynamical Solution

When a complete analysis of the electrical response of the material isrequired either for reasons of highest precision or when the form of theZ_(S) (ω) is more complex and the detailed analysis of the boundaries ofthe sample (interface element R_(M+S) and C_(M+S)) is needed, the fulldynamical solution of the problem has to be undertaken numerically. Thissituation corresponds to FIG. 4a and no reticulation of the sample intoindividual volume elements is performed (reticulation is not relevant inthis case since the reticulation of the sample and the static R,Cnetwork analysis of the electrical response of the sample is theapproximative solution representation of the full dynamical solution ofthe problem).

Besides the governing (transport) equations for the local charge densityρ(x,t) space-time evolution describing the electrical response of thesample to the external voltage input, it is necessary to define (tomodel) the initial condition for the ρ(x,t) and the appropriate boundaryconditions (transport of the electrical charge across theboundaries--surface of the sample).

Once the initial and the boundary conditions are defined the set of thegoverning equations for the space-time evolution of the total, localcharge density ρ(x,t) is solved numerically with some starting values ofthe four electrical material parameters Δμ^(ch), ε, N and μ. Theresulting calculated electrical response function Z_(CAL) (ω) of thesample is then compared with the experimentally observed complexelectrical impedance Z_(EXP) (ω) and the whole process undergoesiterations until the required agreement between Z_(CAL) (ω) and Z_(EXP)(ω) is obtained, leading to the final values of the four electricalmaterial parameters Δμ^(ch), ε, N and μ characterizing the electricalproperties of the material under investigation.

At this level of solution the computer is essential since the problem ofcalculating Z_(CAL) (ω) can be done only numerically.

Verification of the Validity of the Present Invention ExperimentalExamples

The first set of experimental tests has been performed with pure,monocrystalline silicon samples of both types (n-type and p-type) withvarying degree of doping using either evaporated thick gold film or aconductive silver paint as metal electrode material.

Typical raw experimental results of the electrical response in thesesamples are shown in FIGS. 6a and 6b where log {Re (Z_(S) (ω))} and log{Im (Y_(S) (ω)/ω)} are plotted versus log of the measurement frequencyfor a number of different temperatures.

The verification of the validity of the present invention has beenproved in two different ways. The first is demonstrated in the Table 2where the numerical values of the four electrical material parametersΔμ^(ch), ε, N and μ are compared either with the literature values(Δμ^(ch), ε and μ) or with the values obtained by using 4PDCR andFTIR/FTIPL experimental methods on the same samples.

A high degree of agreement has been obtained (to within 5%) limited onlyby the precision of the determination of the geometrical dimensions ofthe sample.

As is apparent from the Table 2 the values of the four electricalmaterial parameters determined by the "New" EIS method (presentinvention) are more precise than the values for these parametersobtained by the other methods or obtained from the literature.

The second way of verifying (testing) the validity of the presentinvention was the use of the temperature variations of theexperimentally observed Z_(S) (ω) of the samples (temperature being usedas a known parameter).

The four electrical material parameters have been extracted from themeasured Z_(S) (ω) at a given temperature by fitting these fourparameters at this single temperature. The form and the numerical valuesof the Z_(S) (ω, T) were then calculated at other chosen temperatures(the temperature dependence of the Z_(S) (ω, T) is known--it is also oneof the results of the theoretical analysis of the electrical responseembodied in the present invention). The calculated values for Z_(S) (ω,T) were then compared with the observed Z_(S) (ω, T) at chosentemperatures over the entire frequency range measured in the experiment.A complete agreement has been found proving beyond reasonable doubt thevalidity of the present invention.

The second set of experimental tests has been performed with pure,polycrystalline silicon of high purity, without the prior knowledge ofthe type of the majority, mobile electrical charge carriers. Evaporated,thick gold film or conductive silver paint were used as metal electrodematerial.

The typical raw experimental results of the electrical response in thesesamples, are shown in FIGS. 7a and 7b, where log {Re[Z_(s) (ω)]} and log{Im[Y_(s) (ω)]} are plotted versus log of the measurement frequency fora number of different temperatures.

Using the present invention it was possible to identify the electricaltopology of silicon grains in polycrystalline silicon samples (FIG. 8a),the nature of the electrical transport in polycrystalline silicon, theelectrical characteristics of the individual grains/grain boundaries(FIGS. 8b, 8c) and the concentration N of the majority mobile chargecarriers (electrical purity) in these polycrystalline silicon samples.

With N as free parameter an extremely good numerical fit to the rawexperimental data (FIGS. 7a, 7b) at different temperatures was obtained,using the static R,C electrical network analysis (FIGS. 8a, 8b, 8c) ofthe present invention. The best numerical value for the quantity N wasfound to be approximately 6·10¹⁸ m⁻³ indicating an extreme purity of thestarting polycrystaline silicon material.

Comparison between the new EIS experimental method according to theinvention and the prior C-V (Quasistatic, d.c. voltage bias dependentCapacitance measurement) and DLTS (Deep Level Transient Spectroscopy)experimental methods.

At this stage it is appropriate to compare at least qualitatively thenew EIS experimental method according to the present invention with twoother experimental methods and variations thereof that are used todayfor the electrical characterization of semiconducting materials. It isclaimed that they are capable of yielding the values of N and/or N(x)(the density of the majority mobile charge carriers; C-V experimentalmethods) and the energy position and the density of the deep localizedenergy levels within the gap of a measured semiconductor (the quantitiesN⁺ _(i),LOC (E_(i), x) and N⁻ _(j),LOC (E_(j), x) in equation system(1); DLTS experimental methods).

Both of these methods attempt to measure the depletion capacitance C_(D)(equation (5) and FIG. (5)) and its changes either with an externallyapplied d.c voltage bias V_(d).c.^(BIAS) (C-V methods) or withtemperature T (DLTS methods).

However, they both suffer from a fundamental physical flaw in that themeasuring frequency e at which the wanted capacitance C_(D) of thesystem under investigation is to be measured is chosen arbitrarily. Aquick inspection of FIG. 5 shows that if this arbitrarily chosenfrequency used by the two methods does not fall within a narrow rangearound f_(D) (flat part of C₂)) both methods will yield incorrectresults (the measured capacitance not related to C_(D)) with deducedelectrical material parameters that bear no relation to their truevalues. This is particularly so for the higher resistivitysemiconductors and insulators where the usual frequency used (1MHz)semiconductors and insulators where the usual frequency used (1MHz)would yield a measurement of the bulk capacitance C_(B) rather than thatof the depletion capacitance C_(D) (equation (5) and FIG. 5).

It is also evident that when the measuring frequency is chosen correctly(close to the flat part of the C₂ --FIG. 5), both methods arerepresented by a single point on the measured electrical impedance curveZ (ω) (Im(Y(ω)/ω)) of the present invention. In such case they thereforerepresent a measurement of the level C₂ (FIG. 5) as a function of thed.c. voltage bias and/or as a function of the temperature T.

The knowledge and the proper analysis of the full Z (9) curve (presentinvention) is however essential in order to decide about the validity ofthe C-V and DLTS results.

It should also be pointed out that the EIS experimental method accordingto the present invention measures N and/or N(x) directly through themeasurement of the Z (ω) over the entire relevant frequency range. Oneof the major results of the present invention is namely the discovery ofone to one correspondanco between the measuring frequency ω and thespatial co-ordinate x. The measurement of Z (ω) at frequency ω_(i) is ameasure of the density of the mobile electrical charges N(x) at pointx_(i). This is a result of the solution of the Poisson equation and itis valid under small signal approximation--the usual condition of theEIS experiment.

In conclusion it should be stressed again that the C-V and DLTS type ofmeasurements are embodied automatically (as a single point in the Z (ω)curves) in the EIS experimental method according to the presentinvention. No dedicated C-V or DLTS apparatus is necessary and bymeasuring Z (9) in the whole frequency range there is a quarantee thatno errors arise from incorrectly chosen measuring frequency.

It is therefore desired that the invention not be limited to thepreferred embodiment (FIGS. 1 to 8) and it is intended to cover in theappended claims all such variations or modifications as fall within thescope of the present invention.

REFERENCES

(1) S. M. Sze (1981) "Physics of Semiconductor Devices" (John Wiley &Sons)

(2) E. H. Nicollian and I. R. Brews (1982) "MOS (Metal OxideSemiconductor) Physics and Technology" (John Wiley & Sons)

(3) I. Ross MacDonald editor (1987) "Impedance Spectroscopy" (John Wiley& Sons)

(4) R. C. Weast editor (1974) "Handbook of Chemistry and Physics" (CRCPress, Cleveland, Ohio)

                                      TABLE 1                                     __________________________________________________________________________    EXP Method                                                                           ρ(ΩM)                                                                   ε(FM.sup.-1)                                                               μ(M.sup.2 v.sup.-1 s.sup.-1)                                                      N(M.sup.-3)                                                                        Δμ.sup.ch (eV)                                                              Comment                                    __________________________________________________________________________    Standard EIS                                                                             Yes                     Only if properly analysed                  Standard EIS                                                                         Yes                         Only if properly analysed                  4 PDCR Yes                         Only for ρ ≦ 1kΩ cm        4 PDCR          Yes    Yes         Only in combination with Hall Effect      He              Yes    Yes         and for ρ ≦ 1kΩ cm                                           (unusable for Glasses)                      FTIR                   Yes         Only in monocrystalline                                                      semiconductors                             FTIPL                  Yes         Measurements at liq.He temperatures)        FTIR   Yes                         Only if μ known                        FTIPL  Yes                         (Measurements at lig.He temperatures)      TOF             Yes                Problems with thin resistive samples       CP                          Yes                                               TEE                         Yes    Large Discrepancies                        PEE                         Yes    among the measured values                  MAR    Yes Yes                     Only if properly Analysed                  "NEW" EIS                                                                            Yes Yes  Yes    Yes  Yes                                               __________________________________________________________________________     Four Probe d.c. electrical resistivity (4PDCR)                                Standard Electrical Impedance Spectroscopy (EIS)                              Hall Effect (HE)                                                              Fast Fourier Transform Infrared Photoluminescence (FTIPL)                     Fast Fourier Transform Infrared spectroscopy (FTIR)                           Contact Potential (CP)                                                        Thermal Electron Emission (TEE)                                               Electrical Time  of  Flight (TOF)                                             Photoelectrical Electron Emission (PEE)                                       Microwave AbsorptionReflection (MAR)                                     

                                      TABLE 2                                     __________________________________________________________________________    SAM-                                                                          PLE MAJORITY                                                                             ELEC- EIS   FTIR/FTIPL                                                                            EIS  4PDCR                                                                              EIS                                                                              Literature                                                                          EIS   Literature            n.  CARRIERS                                                                             TRODES                                                                              N(M.sup.-3)                                                                         N(M.sup.-3)                                                                           ρ(Ωcm)                                                                   ρ(Ωcm)                                                                   ε/ε.sub.O                                                        ε/ε.sub.O                                                           Δμ.sup.ch                                                                  Δμ.sup.ch                                                             (eV)                 __________________________________________________________________________    n.1 n      Au    1.90 · 10.sup.17                                                           2.98 · 10.sup.17                                                                       12.0                                                                             11.9  0.6 ± 0.05               n.94                                                                              n      Au    4.02 · 10.sup.17                                                           2.98 · 10.sup.17                                                                       12.0                                                                             11.9  0.6                         TOP3                                                                              n      Au                   782  855 12.0                                                                             11.9  0.6                         TOP3                                                                              n      Ag                   782  855 12.0                                                                             11.9  0.6                         n.19                                                                              p      Au    6.63 · 10.sup.17                                                           7.44 · 10.sup.17                                                                       12.0                                                                             11.9  0.4 ± 0.10                                                                         0.8 + 0.4           n.19                                                                              p      Ag    6.63 · 10.sup.17                                                           7.44 · 10.sup.17                                                                       12.0                                                                             11.9  0.4                         n.92                                                                              p      Au    7.55 · 10.sup.17                                                           10.5 · 10.sup.17                                                                       12.0                                                                             11.9  0.4                         TOP1                                                                              p      Au                  8741 11480                                                                              12.0                                                                             11.9  0.4                         TOP1                                                                              p      Ag                  8741 11480                                                                              12.0                                                                             11.9  0.4                         __________________________________________________________________________

We claim:
 1. A method for determining characteristic electricalproperties of semi-conducting materials wherein the time or frequencydependent electrical impedance Z(t) or Z(ω), respectively, or admittanceY(t) or Y(ω), respectively, is measured, comprising the steps ofestablishing a finite, positive (n-type material) or negative (p-typematerial) difference in the chemical potential between the inner and thesurface of the material or between the inner of the material and themetal electrode--material surface interface layer, thereby causing theformation of depletion regions near the electrical contacts and by usingthe effect of this established difference in the chemical potential onthe measured electrical impedance or admittance the step of determiningfrom this measured electrical impedance or admittance one, more or allof the following electrical parameters f the material:the differenceΔμ^(ch) in chemical potential, the dielectric constant ε, the density Nof majority mobile charge carriers, the density N_(min) of minoritymobile charge carriers, the electrical mobility μ of majority mobilecharge carriers, the electrical mobility μ_(min) of minority mobilecharge carriers, the emission and capture rates E.R and C.R,respectively, for mobile positive and negative charge carrierscharacterizing the effect of surface and bulk localized states withinthe band gap (Eg), when they are present, on the electrical transportand the measured electrical impedance, whereby the determination of theelectrical parameters of the material is carried out by solving a systemof equations for the total charge density ρ(x,t) consisting of themobile negative and positive charge densities and localized negative andpositive charge densities in the material supplemented with initial andboundary conditions, where the space-time development of ρ(x,t)determines the electric current running in the external circuit inresponse to the applied electric voltage thereby defining the electricalcomplex impedance Z_(s) (ω) of the material.
 2. The method according toclaim 1, wherein the electrical parameters of the metalelectrode-material surface interface layer such as the emission andcapture cross sections σ_(e) and σ_(c) of the interface layer for themobile positive and negative charge carriers are determined from themeasured electrical impedance by means of the initial and boundaryconditions for ρ(x,t).
 3. The method according to claim 1, in which theinterface layer is a Si-SiO₂ -metal electrode interface layer, whereinthe electrical elements R_(M+S) and C_(M+S) of the interface layer aredetermined by means of the expressions: ##EQU7## and wherein thequantities appearing in the analytical expressions have the followingmeaning:εSiO₂ --dielectric constant of the silicon dioxide dSiO₂--thickless of the native silicon dioxide layer A_(r) --total area ofthe sampleRichardson constant ΔE₁ --quantum runnelling barrier heightm--electron rest mass h--Pfanck's constant k--Boltzmann's constantT--absolute temperature ΔE₂ --activation energy |E|--absolute value ofelemental charge.
 4. The method according to claim 1, wherein that theelectrical parameters of the material are determined from the followingexpressions for capacitance and resistance of the inner of the materialand the depletion region thereof, respectively: ##EQU8## which areidentifiable as levels on the measured electrical impedance curve Z (ω),and where the quantities appearing in the analytical expressions havethe following meaning:μ(T)--electrical mobility of The mobile, majorityel. charge carriers μ_(c) (T)--electrical mobility of the electrons atE_(c) μ_(n) (T)--electrical mobility of the holes at E_(v) N--mobile,majority el. charge carriers particle number density L--total length ofthe sample W_(o) (T)--depletion width of the sample A_(r) --total,active area of the sample C_(o) ·ε_(r) =C--dielectric constant of thesample ε_(q) --energy gap of the material Δμ^(ch) (T)--difference in thechemical potential N_(c), N_(v) --one particle effective densities ofstates in the conduction (E_(c)) and valence (E_(v)) bands respectively|E|--elemental electrical charge k--Boltzmann constant T--absolutetemperature E_(O) ^(P) --Energy difference between the top of thevalence band and the electrochemical potential at the surface of thesample E_(O) ^(M) --Energy difference between the bottom of theconduction band and the electrochemical potential at the surface of thesample Z--variable in the function arctg having integration limits u and1.0.
 5. The method according to claim 1, wherein the electricalparameters of the material are determined from the capacitance andresistance of the interface layer and the inner of the material and theelectrical impedance of the depletion region by the followingexpressions: ##EQU9## and wherein the quantities appearing in theanalytical expressions have the following meaning:εSiO₂ --dielectricconstant of the silicon dioxide dSiO₂ --thickness of the native silicondioxide layer A_(r) --total area of the sample ARichardson constant ΔE₁--quantum runnelling barrier height m--electron rest mass h--Planck'sconstant k--Bolzmann's constant T--absolute temperature ΔE₂ --activationenergy |E|--absolute value of elemental charge, ##EQU10## and whereμ_(e) (T)--electrical mobility of the electrons at E_(c) μ_(n)(T)--electrical mobility of the holes at E_(v) N_(c), N_(v)--one-particle effective densities of states in the conduction (E_(c))and valence (E_(v)) bands ε_(o) ·ε_(r) =ε--dielectric constant of thesample |e|--elemental electrical charge E_(g) --energy gap of thematerial k--Boltzmann constant T--absolute temperature W_(D)(T)--depletion width of the sample Δμ^(ch) (T)--difference in thechemical potential N--mobile, majority electrical charges particlenumber density A_(r) --total, active area of the sample,and ##EQU11##|e|--elemental electrical change μ(T)--electrical mobility of themobile, majority electrical charge carriers, N--mobile electrical chargecarriers particle number density W_(D) (T)--depletion width of thesample Δμ^(ch) (T)--difference in chemical potential ε_(o) ·ε_(r)=ε--dielectric constant of the sample L--total length of the sampleAr--total active area of the sampleand where the values of C_(M+S) andR_(M+S) are identifiable as levels on the measured impedance curve Z(ω)if such levels are present.
 6. The method according to claim 1, whereinfor the "research" experimental set-up the material is depositedsemipermanently on the metal electrodes or vice versa.
 7. The methodaccording to claim 1, wherein for "quality control" set-up the metalelectrodes are brought into contact with the surface of the material viaan electrically conducting medium which ensures a well-definedelectrical contact area on the specimen, such as an elastic or flowablematerial which may be brought into and out of engagement with one orboth surfaces of the material.
 8. The method according to claim 7,wherein the measurement is repeated in points distributed throughout thearea of the specimen for mapping of the electrical parameters of thematerial.
 9. An apparatus for carrying out the method according to claim1, comprising an external electrical circuit for measuring the time orfrequency dependent electrical impedance Z(t) or Z(ω) or admittance Y(t)or Y(ω) of the material and means for electrically contacting thesurface or surfaces of the material, means for providing a finite,positive (n-type material) or negative (p-type material) difference inthe chemical potential between the inner and the surface of the materialor between the inner of the material and the metal electrode--materialsurface interface layer, thereby causing the formation of depletionregions near the electrical contacts and means for determining from themeasured electrical impedance or admittance one, more or all of thefollowing electrical parameters of the material:the difference Δμ^(ch)in chemical potential, the dielectric constant ε, the density N ofmajority mobile charge carriers, the density N_(min) of minority mobilecharge carriers, the electrical mobility μ of majority mobile chargecarriers, the electrical mobility μ_(min) of minority mobile chargecarriers, the emission and capture rates E.R and C.R, respectively, ofmobile positive and negative charge carriers characterizing the effectof surface and bulk localized states within the band gap, when they arepresent, on the electrical transport and the measured electricalimpedance.
 10. The apparatus according to claim 9, wherein it isarranged for measuring the electrical impedance/admittance in afrequency range from 1 mHz to 10 GHz (1000 s-0.10 ns) and with a realcomponent of the impedance in the range from 0,00I ohm to 10¹⁴ ohm andan imaginary component of the admittance divided by the frequency ω inthe range from 10⁻¹⁴ Farad to 10⁻² Farad.
 11. The apparatus according toclaim 9, comprising a specimen support having a plurality of coaxialleads for connection to the specimen of the external electricalmeasurement circuit and means for controlling the temperature of thespecimen.
 12. The apparatus according to claim 9, comprisingelectrically conducting contact means providing for a well definedelectrical contact area on the specimen and which are arranged for beingbrought into and out of engagement with one or both of the surfaces ofthe material, and control means for relatively adjusting the contactmeans and the specimen throughout the total area thereof.